Normal (mathematics): Difference between revisions

Latest revision as of 07:02, 26 December 2008

The term normal in mathematics is used in the following broad senses:

To denote something upright or perpendicular
To denote something that is as it should be. In this sense, normal means good or desirable rather than typical

It is typically used as a limiting adjective, and in a binary sense. In other words, an object in a certain class is normal if it satisfies certain conditions, and every object in that class is either normal or not normal. The noun form is normality. Thus, normality can be viewed as a property in various contexts.

In group theory

Normal subgroup: A subgroup of a group that occurs as the kernel of a homomorphism, or equivalently, such that every left coset and right coset are equal.

Related terms: Normality (the property of a subgroup being normal), normal core (the largest normal subgroup contained in a given subgroup), normal closure (the smallest normal subgroup containing a given subgroup), normalizer (the largest subgroup containing a given subgroup, in which it is normal), normal automorphism (an automorphism that restricts to an automorphism on every normal subgroup).

Term variations: subnormal subgroup, abnormal subgroup, quasinormal subgroup, and others. See Groupprops:Category:Variations of normality, Groupprops:Category:Opposites of normality, and Groupprops:Category:Analogues of normality.

Primary subject wiki entry: Groupprops:Normal subgroup. See also Groupprops:Questions:Normal subgroup.

Other subject wiki entries: Diffgeom:Normal subgroup

Coverage in guided tours: Groupprops guided tour for beginners; section not yet prepared.

Also located at: Wikipedia:Normal subgroup, Planetmath:NormalSubgroup, Mathworld:NormalSubgroup, Springer Online Reference Works, Citizendium:Normal subgroup

In topology

Normal space: A topological space is termed normal if all points are closed sets (the $T_{1}$ assumption), and any two disjoint closed subsets can be separated by disjoint open subsets.

In some definitions, the $T_{1}$ assumption is skipped.

Related terms: normality (the property of a topological space being normal)

Term variations: Variations of normality offers a list.

Primary subject wiki entry: Topospaces:Normal space

Also located at: Wikipedia:Normal space, Planetmath:NormalSpace, Mathworld:NormalSpace

In linear algebra

Normal form: also called canonical form, is a standard form in which to put a matrix (typically, upto conjugation). Two normal forms commonly used are the Jordan normal form and the rational normal form.

Normal matrix: A matrix over the complex numbers, that commutes with its conjugate-transpose.

No primary subject wiki entry.

Also located at: Wikipedia:Normal matrix, Mathworld:NormalMatrix, Planetmath:NormalMatrix

Normal operator

In field theory

Normal field extension: A field extension is termed normal if the fixed field under the automorphism group of the extension, is precisely the base field.

Primary subject wiki entry: Galois:Normal extension

Also located at: Planetmath:NormalExtension, Wikipedia:Normal extension, Mathworld:NormalExtension

In commutative algebra

Normal domain: An integral domain is termed normal if it is integrally closed in its field of fractions.

Primary subject wiki entry: Commalg:Normal domain

Normal ring: A commutative unital ring is termed normal if it is integrally closed in its total quotient ring.

Primary subject wiki entry: Commalg:Normal ring

In differential geometry

Normal to a curve in the plane: The line perpendicular to the tangent to the curve at a point.

Normal bundle to a submanifold, or immersed manifold: The quotient of the tangent bundle to the whole manifold (restricted to the submanifold), by the tangent bundle to the submanifold, or immersed manifold. When the manifolds are Riemannian, the normal bundle can be viewed as the orthogonal complement to the tangent bundle of the submanifold, in the tangent bundle to the whole manifold.

Vectors in the normal bundle are termed normal vectors.

Main subject wiki entry: Diffgeom:Normal bundle

Also located at: Wikipedia:Normal bundle, Mathworld:NormalBundle, Planetmath:NormalBundle

Normal coordinate system

In category theory

Normal monomorphism: A monomorphism in a preadditive category, or more generally, in a category enriched over pointed sets, that arises as the kernel of an epimorphism.

Primary subject wiki entry: Cattheory:Normal monomorphism

Normal epimorphism: An epimorphism in a preadditive category, or more generally a category enriched over pointed sets, that arises as the cokernel of a monomorphism.

Primary subject wiki entry: Cattheory:Normal epimorphism

Normal category: A preadditive category, or more generally a category enriched over pointed sets, in which every monomorphism is a normal monomorphism.

Primary subject wiki entry: Cattheory:Normal category

@@ Line 3: / Line 3: @@
 * To denote something upright or perpendicular
 * To denote something that is as it ''should be''. In this sense, normal means ''good'' or ''desirable'' rather than ''typical''
+It is typically used as a limiting adjective, and in a binary sense. In other words, an object in a certain class is ''normal'' if it satisfies certain conditions, and every object in that class is either normal or not normal. The noun form is ''normality''. Thus, normality can be viewed as a property in various contexts.
 ===In group theory===
-[[Groupprops:Normal subgroup|Normal subgroup]]: A subgroup of a group that occurs as the kernel of a homomorphism, or equivalently, such that every left coset and right coset are equal.
+{{:Normal subgroup}}
-Related terms: Normality (the property of a subgroup being normal), [[Groupprops:Normal core|normal core]] (the largest normal subgroup contained in a given subgroup), [[Groupprops:Normal closure|normal closure]] (the smallest normal subgroup containing a given subgroup), [[Groupprops:Normalizer|normalizer]] (the largest subgroup containing a given subgroup, in which it is normal).
-Term variations: [[Groupprops:Subnormal subgroup|subnormal subgroup]], [[Groupprops:Abnormal subgroup|abnormal subgroup]], [[Groupprops:Quasinormal subgroup|quasinormal subgroup]], and others. See [[Groupprops:Category:Variations of normality]], [[Groupprops:Category:Opposites of normality]], and [[Groupprops:Category:Analogues of normality]].
 ===In topology===
-[[Topospaces:Normal space|Normal space]]: A topological space is termed normal if all points are closed sets (the <math>T_1</math> assumption), and any two disjoint closed subsets can be separated by disjoint open subsets.
+{{:Normal space}}
-In some definitions, the <math>T_1</math> assumption is skipped.
-Related terms: normality (the property of a topological space being normal)
-Term variations: [[Topospaces:Category:Variations of normality|Variations of normality]] offers a list.
 ===In linear algebra===
-[[Find link::Normal form]], also called canonical form, is a standard form in which to put a matrix (typically, upto conjugation). Two normal forms commonly used are the Jordan normal form and the rational normal form.
+{{:Normal form (linear algebra)}}
-[[Find link::Normal matrix]]: A matrix over the complex numbers, with the property that it commutes with its conjugate-transpose.
+{{:Normal matrix}}
-===In Galois theory===
-[[Find link::Normal field extension]]: A field extension is termed normal if the fixed field under the automorphism group of the extension, is precisely the base field.
+{{:Normal operator}}
+===In field theory===
+{{:Normal field extension}}
 ===In commutative algebra===
-[[Commalg:Normal domain|Normal domain]]: An integral domain is termed normal if it is integrally closed in its field of fractions.
+{{:Normal domain}}
-[[Commalg:Normal ring|Normal ring]]: A commutative unital ring is termed normal if it is integrally closed in its total quotient ring
+{{:Normal ring}}
 ===In differential geometry===
@@ Line 41: / Line 34: @@
 Normal to a curve in the plane: The line perpendicular to the tangent to the curve at a point.
-[[Diffgeom:Normal bundle|Normal bundle]] to a submanifold, or immersed manifold: The quotient of the tangent bundle to the whole manifold (restricted to the submanifold), by the tangent bundle to the submanifold, or immersed manifold. When the manifolds are Riemannian, the normal bundle can be viewed as the orthogonal complement to the tangent bundle of the submanifold, in the tangent bundle to the whole manifold.
+{{:Normal bundle}}
+{{:Normal coordinate system}}
-[[Diffgeom:Normal coordinate system|Normal coordinate system]] refers to a particular kind of local coordinate system for a Riemannian manifold.
+===In category theory===
-Vectors in the normal bundle are termed normal vectors.
+{{:Normal monomorphism}}
-===In normed vector spaces===
+{{:Normal epimorphism}}
-In general normed vector spaces, as well as in tangent spaces to Riemannian manifolds, ''normalize'' means to scale by a factor to make the norm equal to 1.
+{{:Normal category}}
 ===In probability/statistics===
-[[Find link::Normal distribution]]: A particular kind of distribution parametrized by mean and variance, also called the Gaussian distribution.
+{{:Normal distribution}}
+===In axiomatic set theory===
-[[Find link::Normal random variable]]: A random variable whose distribution is a normal distribution.
+{{:Normal function}}
+{{:Normal measure}}
 ===Others===
-[[mathworld:NormalNumber|Normal number]] to base <math>b</math>: A real number that, when written in base <math>b</math>, has all digits occurring with equal probability.
+{{:Normal number}}
-[[mathworld:AbsolutelyNormal|Absolutely normal number]]: A real number that is positive to base <math>b</math> for every integer <math>b > 1</math>.